Chapter 6: Q. 86 (page 411)

In Problem $85 a n d 86$ graph each of the function
$g (x) = \{\begin{cases} 2 \sin x i f 0 \leq x \leq π \\ \cos x + 1 i f π < x \leq 2 π \end{cases}$

Short Answer

Expert verified

Graph of the given function

Step by step solution

Step 1.Given information

The given function $g (x) = \{\begin{cases} 2 \sin x i f 0 \leq x \leq π \\ \cos x + 1 i f π < x \leq 2 π \end{cases}$

Now graph each function as follows.
The given function is a composite function of function $y = 2 \sin x$ for the interval $0 \leq x \leq π$

and function $y = \cos x + 1$ for the interval $π < x \leq 2 π$ .

Here domain of the function $g (x)$ is $[0, 2 π]$ and its range is 1 to - 1.

Step 2.Calculate some points for sinx in their given domain

Choose different x-values and calculate the corresponding $y = \sin x$ values.

x	y=2sinx	(x,y)
0	$2 \sin 0 = 0$	$(0, 0)$
$\frac{π}{4}$	$2 \sin (\frac{π}{4}) = 2 \frac{\sqrt{2}}{2} = \sqrt{2}$	$(\frac{π}{4}, \sqrt{2})$
$\frac{π}{2}$	$2 \sin (\frac{π}{2}) = 2 (1) = 2$	$(\frac{π}{2}, 2)$
$\frac{3 π}{4}$	$2 \sin (\frac{3 π}{4}) = 2 \frac{\sqrt{2}}{2} = \sqrt{2}$	$(\frac{3 π}{4}, \sqrt{2})$
$π$	$2 \sin (π) = 0$	$(π, 0)$

Step 3.Calculate some points for cosx in their given domain

Choose different-xvalues and calculate the

y = \cos x + 1

corresponding values.

$x$	$y = \cos x + 1$	$(x, y)$
$\frac{5 π}{4}$	$\cos (\frac{5 π}{4}) + 1 = - \frac{\sqrt{2}}{2} + 1 = 0.293$	$(\frac{5 π}{4}, 0.293)$
role="math" localid="1646328747587" $\frac{3 π}{2}$	$\cos (\frac{3 π}{2}) + 1 = 0 + 1 = 1$	$(\frac{3 π}{2}, 1)$
$\frac{7 π}{4}$	$\cos (\frac{7 π}{4}) = \frac{\sqrt{2}}{2} + 1 = 1.7071$	$(\frac{7 π}{4}, 1.7071)$
$2 π$	$\cos (2 π) + 1 = 1 + 1 = 2$	$(2 π, 2)$

Step 4.Plot the above ordered pairs on the graph and connect them with a smooth curve.

Thus function $g (x)$ is graphed as a composite function of $y = 2 \sin x$ in blue and $y = \cos x + 1$ in pink shown below:

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In Problem $85 a n d 86$ graph each of the function
$g (x) = \{\begin{cases} 2 \sin x i f 0 \leq x \leq π \\ \cos x + 1 i f π < x \leq 2 π \end{cases}$

Short Answer

Step by step solution

Step 1.Given information

Step 2.Calculate some points for sinx in their given domain

Step 3.Calculate some points for cosx in their given domain

Step 4.Plot the above ordered pairs on the graph and connect them with a smooth curve.

One App. One Place for Learning.

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In Problem 85and86graph each of the functiong(x)=2sinxif0≤x≤πcosx+1ifπ<x≤2π

Short Answer

Step by step solution

Step 1.Given information

Step 2.Calculate some points for sinx in their given domain

Step 3.Calculate some points for cosx in their given domain

Step 4.Plot the above ordered pairs on the graph and connect them with a smooth curve.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Decision Maths

Calculus

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Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problem $85 a n d 86$ graph each of the function
$g (x) = \{\begin{cases} 2 \sin x i f 0 \leq x \leq π \\ \cos x + 1 i f π < x \leq 2 π \end{cases}$